RIGOROUS RESULTS ON THE THERMODYNAMICS OF THE DILUTE HOPFIELD MODEL

We study the Hopfield model of an autoassociative memory on a random graph on N vertices where the probability of two vertices being joined by a link is p(N). Assuming that p(N) goes to zero more slowly than O(1/N), we prove the following results: (1) If the number of stored patterns m(N) is small enough such that m(N)/Np(N) down 0, as N up infinity, then the free energy of this model converges, upon proper rescaling, to that of the standard Curie-Weiss model, for almost all choices of the random graph and the random patterns. (2) If in addition m(N) < In N/ln 2, we prove that there exists, for T < 1, a Gibbs measure associated to each original pattern, whereas for higher temperatures the Gibbs measure is unique. The basic technical result in the proofs is a uniform bound on the difference between the Hamiltonian on a random graph and its mean value.